In conclusion, the microscopic properties of the superconducting state are investigated for the low carrier, quasi-skutterudite Lu\(_3\) Os\(_4\) Ge\(_{13}\) . The ZF-\(\mu \) SR data indicate spontaneous time-reversal-symmetry breaking in the superconducting state, reflected by the increase in relaxation rate \(\Lambda \)  in Fig. REF (c) at a temperature, \(T_{\text{onset}} \simeq \)  2.2K, below the superconducting transition temperature \(T_c \simeq 3.1\,\textrm {K}\) . The measured strength of the dilute magnetic field in the superconducting state of Lu\(_3\) Os\(_4\) Ge\(_{13}\)  is \(\simeq \)  0.18mT—a value much larger than that of most TRSB superconductors, while only those of PrOs\(_4\) Sb\(_{12}\)  and Re\(_2\) Hf are comparable yet still smaller [1]}, [2]}.
The variation of \(\sigma _{sc}\)  with temperature, extracted from our TF measurements, is inconsistent with a single isotropic pairing state. Meanwhile, it is well described by a two-gap scenario, with large gap anisotropy \(\Delta _{0,1}/\Delta _{0,2} \simeq 5-12\) , in accordance with previous studies [3]}, [4]} and with the presence of multiple bands at the Fermi level [3]}; we emphasize that our data is consistent with both fully gapped and nodal states, see Table REF  and Fig. REF (c).
A symmetry analysis of pairing states shows that reaching a TRSB superconducting state in Lu\(_3\) Os\(_4\) Ge\(_{13}\)  via a single continuous phase transition is only consistent with the latter, nodal-pairing scenario. However, a full gapped, TRSB state could be naturally reached via two consecutive transitions.
